The number of solutions of the equation \(5{x^2} + 7x - 4\) is \(\boxed2\). Option (c) is correct.
Further explanation:
The Fundamental Theorem of Algebra states that the polynomial has n roots if the degree of the polynomial is n.
\(f\left( x \right) = a{x^n} + b{x^{n - 1}} + \ldots + cx + d\)
The polynomial function has n roots or zeroes.
Given:
The equation is \(f\left( x \right) = 5{x^2} + 7x - 4.\)
The options are as follows,
(a). 0
(b). 1
(c). 2
(d). 3
Explanation:
The given equation is \(f\left( x \right) = 5{x^2} + 7x - 4.\)
According to the Fundamental Theorem of Algebra the function has 2 roots.
Solve the above equation to obtain the zeros.
\(\begin{aligned}x&= \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\\x &= \frac{{ - 7 \pm \sqrt {{7^2} - 4 \times 5 \times \left( { - 4} \right)} }}{{2 \times 5}}\\x&= \frac{{ - 7 \pm \sqrt {49 + 80} }}{{10}}\\x&= \frac{{ - 7 \pm \sqrt {129} }}{{10}}\\\end{aligned}\)
The number of solutions of the equation \(5{x^2} + 7x - 4\) is \(\boxed2\). Option (c) is correct.
Option (a) is not correct.
Option (b) is not correct.
Option (c) is correct.
Option (d) is not correct.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Polynomial
Keywords: solutions, number of solutions, 0, 1, 2, 3, roots, linear equation, quadratic equation, zeros, function, polynomial, solution, cubic function, degree of the function, multiplicity of 1, multiplicity of 2.
